Deterministic and statistical methods for reconstructing multidimensional NMR spectra

Reconstruction of an image from a set of projections is a well-established science, successfully exploited in X-ray tomography and magnetic resonance imaging. This principle has been adapted to generate multidimensional NMR spectra, with the key difference that, instead of continuous density functions, high-resolution NMR spectra comprise discrete features, relatively sparsely distributed in space. For this reason, a reliable reconstruction can be made from a small number of projections. This speeds the measurements by orders of magnitude compared to the traditional methodology, which explores all evolution space on a Cartesian grid, one step at a time. Speed is of crucial importance for structural investigations of biomolecules such as proteins and for the investigation of time-dependent phenomena. Whereas the recording of a suitable set of projections is a straightforward process, the reconstruction stage can be more problematic. Several practical reconstruction schemes are explored. The deterministic methods-additive back-projection and the lowest-value algorithm-derive the multidimensional spectrum directly from the experimental projections. The statistical search methods include iterative least-squares fitting, maximum entropy, and model-fitting schemes based on Bayesian analysis, particularly the reversible-jump Markov chain Monte Carlo procedure. These competing reconstruction schemes are tested on a set of six projections derived from the three-dimensional 700-MHz HNCO spectrum of a 187-residue protein (HasA) and compared in terms of reliability, absence of artifacts, sensitivity to noise, and speed of computation.